Fractional Precipitation

General Chemistry • Solubility and Complex Ion Equilibria

Written by STEM Calculators Team Published April 18, 2026 Updated April 18, 2026

Fractional Precipitation — Order and Completeness

Compare two competing precipitates formed by the same cation and determine which salt begins to precipitate first. The calculator computes the threshold free cation concentration for each salt, estimates how much of the first ion remains when the second starts to precipitate, and shows interactive graphs only after the results are ready.

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Recognized keys: M, X, Y, a, b, c, d, Ksp1, Ksp2, X0, Y0, precision.

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Species and display labels

Cation symbol M

Anion X

Anion Y

Displayed significant figures

Salt 1 — M_aX_b

a for the cation

b for ion X

K_sp,1

Initial [X]₀ (mol·L^-1)

Salt 2 — M_cY_d

c for the cation

d for ion Y

K_sp,2

Initial [Y]₀ (mol·L^-1)

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What is fractional precipitation?

Fractional precipitation is the selective precipitation of one anion from a mixture when a common cation \(\mathrm{M}^{p+}\) is added. Two sparingly soluble salts can form:

\(\mathrm{M}_{a}\mathrm{X}_{b}(s)\;\rightleftharpoons\; a\,\mathrm{M}^{p+} + b\,\mathrm{X}^{q-}\)

\(\mathrm{M}_{c}\mathrm{Y}_{d}(s)\;\rightleftharpoons\; c\,\mathrm{M}^{p+} + d\,\mathrm{Y}^{r-}\)

With solubility products \[ K_{\mathrm{sp},1} = [\mathrm{M}^{p+}]^{a}[\mathrm{X}^{q-}]^{b},\qquad K_{\mathrm{sp},2} = [\mathrm{M}^{p+}]^{c}[\mathrm{Y}^{r-}]^{d}, \] the salt that needs the lower \([\mathrm{M}^{p+}]\) starts to precipitate first.

Onset (threshold) concentrations

For initial anion concentrations \([\mathrm{X}^{q-}]_0\) and \([\mathrm{Y}^{r-}]_0\) (no depletion yet), the cation thresholds are:

\[ [\mathrm{M}^{p+}]_{\text{th},1} = \Bigg(\frac{K_{\mathrm{sp},1}}{([\mathrm{X}^{q-}]_0)^{b}}\Bigg)^{1/a},\qquad [\mathrm{M}^{p+}]_{\text{th},2} = \Bigg(\frac{K_{\mathrm{sp},2}}{([\mathrm{Y}^{r-}]_0)^{d}}\Bigg)^{1/c}. \]

A useful separation window exists when \[ \min([\mathrm{M}]_{\text{th},1},[\mathrm{M}]_{\text{th},2}) < [\mathrm{M}^{p+}] < \max([\mathrm{M}]_{\text{th},1},[\mathrm{M}]_{\text{th},2}), \] so that only the first salt precipitates.

How much of the first anion remains?

Suppose \(\mathrm{M}_{a}\mathrm{X}_{b}\) precipitates first. When the second salt is just about to form (i.e. at \([\mathrm{M}^{p+}] = [\mathrm{M}^{p+}]_{\text{th},2}\)), the remaining \([\mathrm{X}^{q-}]\) is:

\[ [\mathrm{X}^{q-}]_{\text{remain at 2nd}} = \Bigg(\frac{K_{\mathrm{sp},1}}{([\mathrm{M}^{p+}]_{\text{th},2})^{a}}\Bigg)^{1/b}. \]

The fraction kept is \[ \%\text{ remaining of } \mathrm{X}^{q-} = 100 \times \frac{[\mathrm{X}^{q-}]_{\text{remain at 2nd}}}{[\mathrm{X}^{q-}]_0}. \] (Swap indices \(1 \leftrightarrow 2\) and \(X \leftrightarrow Y\) if the other salt precipitates first.)

How the calculator uses this

Reads \(a,b,c,d\), \(K_{\mathrm{sp},1}\), \(K_{\mathrm{sp},2}\), and \([\mathrm{X}^{q-}]_0,[\mathrm{Y}^{r-}]_0\).
Computes \([\mathrm{M}^{p+}]_{\text{th},1}\) and \([\mathrm{M}^{p+}]_{\text{th},2}\) and reports which salt precipitates first.
Shows the separation window and, at the onset of the second precipitate, the remaining concentration and percent of the first anion.

Assumes dilute solutions, activities ≈ concentrations, no complexation or extra common-ion sources, and negligible volume change on adding the precipitating reagent.

Frequently Asked Questions

What is fractional precipitation in chemistry?

Fractional precipitation is the selective removal of one ion from a mixture by adding a reagent that forms a precipitate. As the precipitating ion concentration increases, the less soluble salt reaches its precipitation threshold first.

How do you decide which salt precipitates first when adding M?

Compute the threshold cation concentrations for each salt and compare them. The salt with the smaller threshold [M]th is the one that begins to precipitate first.

How is the threshold concentration [M]th calculated from Ksp?

For MaXb with initial [X]0, the onset occurs when Ksp1 = [M]^a x [X]^b with [X] approximately equal to [X]0, giving [M]th,1 = (Ksp1 / ([X]0^b))^(1/a). The same approach applies to McYd using Ksp2 and [Y]0.

What does the separation window mean in fractional precipitation?

The separation window is the range of [M] where only the first salt precipitates while the second has not started yet. A larger window generally indicates better selectivity for separating the ions.

What assumptions does this fractional precipitation calculator use?

It assumes dilute solutions where activities are approximated by concentrations, negligible volume change when adding the precipitating reagent, and no complex-ion formation or extra side equilibria. If these effects are significant, predicted thresholds and completeness can differ.

Fractional Precipitation — Order and Completeness

Species and display labels

Salt 1 — MaXb

Salt 2 — McYd

Calculation steps

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Frequently Asked Questions

Related calculators

Salt 1 — M_aX_b

Salt 2 — M_cY_d